LONDON — In a sign of deepening crisis in the Church of England after it rejected the appointment of women as bishops, its spiritual leader said Wednesday that the church had “undoubtedly lost a measure of credibility” and had a “lot of explaining to do” to people who found its deliberations opaque.

The archbishop of Canterbury, the Most Rev. Rowan Williams, was speaking after an emergency meeting of bishops called to debate Tuesday’s narrow balloting by its General Synod rejecting the ordination of women as bishops, even though female priests account for one third of the Church of England’s clergy members. Female priests hold senior positions like canon and archdeacon, and some had been hoping to secure appointments as bishops by 2014 if the change had been approved.

The vote represented a direct rebuff to Williams’ reformist efforts during his 10 years as head of the church and a huge setback to a campaign for change that has been debated intensely and often bitterly for the past decade.

More than 70 percent of the 446 synod votes Tuesday were in favor of opening the church’s episcopacy to women. But the synod’s voting procedures require a two-thirds majority in each of its three “houses”: bishops, clergy and laity. The bishops approved the change by 44-3, and the clergy by 148-45. The vote among the laity, though, was 132-74, six votes less than the two-thirds needed.

The Church of England is the so-called established church, meaning that it is recognized by law as representing the official religion, enjoys special privileges and is supported by the civil authorities.

Some lawmakers suggested Wednesday that the synod vote would create a crisis of church-state relations since the rejection of female bishops contradicted national laws on gender equality. Prime Minister David Cameron, already at loggerheads with the church over the government’s plans to legalize same-sex marriage next year, urged church authorities Wednesday to devise a way out of the impasse.

“I’m very clear the time is right for women bishops; it was right many years ago,” he told Parliament on Wednesday. “They need to get on with it, as it were, and get with the program. But you do have to respect the individual institutions and the way they work while giving them a sharp prod.”

Addressing the synod Wednesday in unusually unambiguous language, Williams declared, “We have, to put it very bluntly, a lot of explaining to do.